Uniqueness of Equilibrium in Steady State and Dynamic Traffic Networks

Michael John Smith, Richard Mounce

Research output: Chapter in Book/Report/Conference proceedingChapter


This paper addresses the issue of uniqueness of equilibrium in traffic networks, which are considered to be directed graphs in which traffic flows along acyclic paths, which are called routes, connecting origin-destination (OD) pairs. The demand for travel between each OD pair is assumed to be rigid. It is shown that in the steady state model, provided that each link cost function is a non-decreasing function of link flow, costs at equilibrium are unique. The paper then goes on to consider dynamic user equilibrium in dynamic traffic models, with special attention given to the bottleneck model. In the dynamic bottleneck queueing model, the route cost vector is not a monotone function of the route flow vector; an example network is given to illustrate this. An alternative definition of what constitutes an increasing function (of a function) is then given; and the cost function is shown to satisfy this condition in the bottleneck model. A number of additional properties are then put forward that must be satisfied in order for the equilibrium flow pattern to be essentially unique in the single OD pair case; the bottleneck model is shown to satisfy these properties.
Original languageEnglish
Title of host publicationTransportation and Traffic Theory 2007
Subtitle of host publicationpapers selected for presentation at ISTTT17, a peer reviewed series since 1959
ISBN (Print) 9780080453750
Publication statusPublished - 2007

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